Dimension estimates for Kakeya sets defined in an axiomatic setting

Abstract

In this dissertation we define a generalization of Kakeya sets in certain metric spaces. Kakeya sets in Euclidean spaces are sets of zero Lebesgue measure containing a segment of length one in every direction. A famous conjecture, known as Kakeya conjecture, states that the Hausdorff dimension of any Kakeya set should equal the dimension of the space. It was proved only in the plane, whereas in higher dimensions both geometric and arithmetic combinatorial methods were used to obtain partial results.In the first part of the thesis we define generalized Kakeya sets in metric spaces satisfying certain axioms. These allow us to prove some lower bounds for the Hausdorff dimension of generalized Kakeya sets using two methods introduced in the Euclidean context by Bourgain and Wolff. With this abstract setup we can deal with many special cases in a unified way, recovering some known results and proving new ones.In the second part we present various applications. We recover some of the known estimates for the classical Kakeya and Nikodym sets and for curved Kakeya sets. Moreover, we prove lower bounds for the dimension of sets containing a segment in a line through every point of a hyperplane and of an (n-1)-rectifiable set. We then show dimension estimates for Furstenberg type sets (already known in the plane) and for the classical Kakeya sets with respect to a metric that is homogeneous under non-isotropic dilations and in which balls are rectangular boxes with sides parallel to the coordinate axis. Finally, we prove lower bounds for the classical bounded Kakeya sets and a natural modification of them in Carnot groups of step two whose second layer has dimension one, such as the Heisenberg group. On the other hand, if the dimension is bigger than one we show that we cannot use this approach.

An improved bound for the dimension of $(\alpha,2\alpha)$-Furstenberg sets

“…Then, by using polar coordinates and Corollary 7.4 in [3], we see that dim H .E/ D dim H .A / D ˛C ˇ. These were the best known bounds prior to this article (see, however, [7,8,13,17] for progress on the corresponding problem in higher dimensions). Note that, because of the min¹ˇ; ˛º term, the bound (1.3) does not distinguish sets in F ˛;˛f rom the (intuitively much larger) sets in F ˛;2˛.…”

Restricting directions for Kakeya sets

“…In [7], Venieri proved that if Ω is d-Alfhors regular then a Ω-Kakeya set X has Hausdorff dimension greater than d+2 2 + 1 2 . Theorem 4 strengthen this result since it gives better estimate for large n and also since we do not make assumption concerning the set of direction Ω.…”

On the dimension of Kakeya sets in the first Heisenberg group

“…The Heisenberg Hausdorff dimension of Euclidean Kakeya sets has been studied in [16] where the author showed a lower bound on the dimension of Kakeya sets. In [17], the author studied Kakeya sets for general metric spaces in axiomatic sense.…”